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This special issue of Studies in Applied Mathematics is dedicated to the 70th birthday of Professor Thanasis Fokas. It contains a selection of papers that all use in some ways the ground‐breaking mathematical ideas and techniques introduced by Thanasis over his long and exceptionally prolific career. [ABSTRACT FROM AUTHOR]

Quantum crystallography is an emerging research field of science that has its origin in the early days of quantum physics and modern crystallography when it was almost immediately envisaged that X‐ray radiation could be somehow exploited to determine the electron distribution of atoms and molecules. Today it can be seen as a composite research area at the intersection of crystallography, quantum chemistry, solid‐state physics, applied mathematics and computer science, with the goal of investigating quantum problems, phenomena and features of the crystalline state. In this article, the state‐of‐the‐art of quantum crystallography will be described by presenting developments and applications of novel techniques that have been introduced in the last 15 years. The focus will be on advances in the framework of multipole model strategies, wavefunction‐/density matrix‐based approaches and quantum chemical topological techniques. Finally, possible future improvements and expansions in the field will be discussed, also considering new emerging experimental and computational technologies. [ABSTRACT FROM AUTHOR]

Recently, the problem of multitasking scheduling has raised a lot of interest in the service industries. Hall et al. (Discrete Applied Mathematics, 2016) proposed a shared processing multitasking scheduling model which allows a team to continue to work on the primary tasks while processing the routinely scheduled activities as they occur. With a team being modeled as a single machine, the processing sharing of the machine is achieved by allocating a fraction of the processing capacity to routine jobs and the remaining fraction, which we denote as sharing ratio, to the primary jobs. In this paper, we generalize this model to parallel machines and allow the fraction of the processing capacity assigned to routine jobs to vary from one to another. The objectives are minimizing makespan and minimizing the total completion time of primary jobs. We show that for both objectives, there is no polynomial time approximation algorithm unless P=NP if the sharing ratios are arbitrary for all machines. Then we consider the problems where the sharing ratios on some machines have a constant lower bound. For each objective, we analyze the performance of the classical scheduling algorithms and their variations and then develop a polynomial time approximation scheme when the number of machines is a constant. [ABSTRACT FROM AUTHOR]

In this paper, we present the general Bernstein functions for the first time. The properties of generalized Bernstein basis functions are given and demonstrated. The classical Bernstein polynomial bases are merely a subset of the general Bernstein functions. Based on the new Bernstein base functions and the collocation method, we present a numerical method for solving linear and nonlinear χ-fractional differential equations (χ-FDEs) with variable coefficients. The fractional derivative used in this work is the χ-Caputo fractional derivative sense (χ-CFD). Combining the Bernstein functions basis and the collocation methods yields the approximation solution of nonlinear differential equations. These base functions can be used to solve many problems in applied mathematics, including calculus of variations, differential equations, optimal control, and integral equations. Furthermore, the convergence of the method is rigorously justified and supported by numerical experiments. [ABSTRACT FROM AUTHOR]

Invasive‐invaded species problems are of relevance in mathematics applied to population dynamics. In this paper, the mentioned dynamics is introduced based on a fourth‐order parabolic operator, together with coupled non‐linear reaction terms. The fourth‐order operator allows us to model a heterogeneous diffusion, as introduced by the Landau–Ginzburg free energy approach. The reaction terms are given by a coupled non‐linear effect in the invasive species, to account for the action of the invaded species and limited resources, and by a non‐Lipschitz term in the invaded species, to account for possible sprouts, once the invasion occurs. The analysis starts by the proof of existence and uniqueness of solutions, making use of the semi‐group theory and a fixed point argument. Asymptotic solutions to the invasive species are explored with an exponential scaling. Afterward, the problem is analyzed with traveling wave profiles, for which a region of positive solutions is explored. [ABSTRACT FROM AUTHOR]

In recent years, reaction–diffusion models have attracted researchers for their wide applications. In this article, we consider Brusselator reaction–diffusion system (BRDS), which is known for its cross diffusion and pattern formations in biology and chemistry. We derive an analytical solution of the fractional Brusselator reaction–diffusion system (FBRDS) with the help of the initial condition by a novel method, residual power series method (RPSM). The system solution has been analyzed by graph. [ABSTRACT FROM AUTHOR]

Clustering is an important tool in statistics, machine learning and applied mathematics. This paper considers the clustering model Y=μlT+Z∈ℝp×n, where the noise matrix Z consists of independent sub‐Gaussian entries Zij and the variance Var(Zij)=σij2 may vary across different coordinates. Our aim is to estimate the error between the label vector l and its defined estimator l^. We provide upper bound estimations for the misclassification rate in the sense of expectation and probability, respectively. Finally, some simulations have been carried out to support our theoretical results and illustrate the advantage of our proposed estimator. [ABSTRACT FROM AUTHOR]

Recently, physics‐informed neural networks (PINNs) have been effectively utilized in a wide range of problems within the domains of applied mathematics and engineering. In PINNs, the governing physical equations are directly incorporated into the loss function of the network and a conventional labeled dataset is not required for its training. In order to successfully simulate the additive manufacturing processes with concrete, a novel process‐based FE‐simulation has been developed where the Drucker–Prager plasticity model is used as the material model. In this work, we will examine the deployment of a PINN to substitute the Newton–Raphson iterations that occur in the return‐mapping algorithm of the Drucker–Prager plasticity model. [ABSTRACT FROM AUTHOR]

We study the problem of prediction of binary sequences with expert advice in the online setting, which is a classic example of online machine learning. We interpret the binary sequence as the price history of a stock, and view the predictor as an investor, which converts the problem into a stock prediction problem. In this framework, an investor, who predicts the daily movements of a stock, and an adversarial market, who controls the stock, play against each other over N turns. The investor combines the predictions of n≥2 experts in order to make a decision about how much to invest at each turn, and aims to minimize their regret with respect to the best‐performing expert at the end of the game. We consider the problem with history‐dependent experts, in which each expert uses the previous d days of history of the market in making their predictions. We prove that the value function for this game, rescaled appropriately, converges as N→∞ at a rate of ON−1/6 to the viscosity solution of a nonlinear degenerate elliptic PDE, which can be understood as the Hamilton‐Jacobi‐Issacs equation for the two‐person game. As a result, we are able to deduce asymptotically optimal strategies for the investor. Our results extend those established by the first author and R.V. Kohn [14] for n=2 experts and d≤4 days of history. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

This is the second part of a two‐paper series that establishes the uniqueness and regularity of a threshold energy wave map that does not scatter in both time directions. Consider the S2‐valued equivariant energy critical wave maps equation on ℝ1+2, with equivariance class k≥4. It is known that every topologically trivial wave map with energy less than twice that of the unique k‐equivariant harmonic map Q→k scatters in both time directions. We study maps with precisely the threshold energy E=2EQ→k. In the first part of the series [15] we gave a refined construction of a threshold wave map that asymptotically decouples into a superposition of two harmonic maps (bubbles), one of which is concentrating in scale. In this paper, we show that this solution is the unique (up to the natural invariances of the equation) two‐bubble wave map. Combined with our earlier work [14] we obtain an exact description of every threshold wave map. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

For a given finite subset S of a compact Riemannian manifold (M,g) whose Schouten curvature tensor belongs to a given cone, we establish a necessary and sufficient condition for the existence and uniqueness of a conformal metric on M\S such that each point of S corresponds to an asymptotically flat end and that the Schouten tensor of the conformal metric belongs to the boundary of the given cone. As a by‐product, we define a purely local notion of Ricci lower bounds for continuous metrics that are conformal to smooth metrics and prove a corresponding volume comparison theorem. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

Artificial intelligence (AI) methods have revolutionized and redefined the landscape of data analysis in business, healthcare, and technology. These methods have innovated the applied mathematics, computer science, and engineering fields and are showing considerable potential for risk science, especially in the disaster risk domain. The disaster risk field has yet to define itself as a necessary application domain for AI implementation by defining how to responsibly balance AI and disaster risk. (1) How is AI being used for disaster risk applications; and how are these applications addressing the principles and assumptions of risk science, (2) What are the benefits of AI being used for risk applications; and what are the benefits of applying risk principles and assumptions for AI‐based applications, (3) What are the synergies between AI and risk science applications, and (4) What are the characteristics of effective use of fundamental risk principles and assumptions for AI‐based applications? This study develops and disseminates an online survey questionnaire that leverages expertise from risk and AI professionals to identify the most important characteristics related to AI and risk, then presents a framework for gauging how AI and disaster risk can be balanced. This study is the first to develop a classification system for applying risk principles for AI‐based applications. This classification contributes to understanding of AI and risk by exploring how AI can be used to manage risk, how AI methods introduce new or additional risk, and whether fundamental risk principles and assumptions are sufficient for AI‐based applications. [ABSTRACT FROM AUTHOR]

We consider the gravity water waves system with a periodic one‐dimensional interface in infinite depth and give a rigorous proof of a conjecture of Dyachenko‐Zakharov [16] concerning the approximate integrability of these equations. More precisely, we prove a rigorous reduction of the water waves equations to its integrable Birkhoff normal form up to order 4. As a consequence, we also obtain a long‐time stability result: periodic perturbations of a flat interface that are initially of size ε remain regular and small up to times of order ε−3. This time scale is expected to be optimal. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

This document is an editorial for a special issue of the journal "Mathematical Methods in the Applied Sciences" focused on William K. Clifford's geometric algebra (GA) and its applications in software engineering, computer graphics, computer vision, and general computer science fields. The editorial highlights the 7th international workshop ENGAGE 2022, which provided a multi-disciplinary approach to GA and resulted in the acceptance of seven papers for this special issue. The editorial also mentions previous conferences and workshops that have emphasized the benefits of GA in computer graphics and vision problems. The special issue includes contributions on various applications of GA, such as protein coordinate prediction, calculation of exponential and elementary functions, octonion Fourier transform, projective duality, and Clifford ratios. The workshop organizers express gratitude to the conference organizers, committee members, reviewers, contributors, and the journal publisher for their support. [Extracted from the article]

This document is an editorial from the journal "Numerical Linear Algebra with Applications" dedicated to celebrating the life and work of Professor Owe Axelsson. Professor Axelsson made significant contributions to numerical analysis, particularly in the development of preconditioning methods and algebraic hierarchical multilevel methods. His work has had a lasting impact on the field and has improved the efficiency of solving complex mathematical problems. The editorial also highlights Professor Axelsson's role as the founder and Editor-in-Chief of the journal and his commitment to establishing it as a high-level scientific journal in Scientific Computing. The special issue includes papers that reflect the ongoing influence of Professor Axelsson's work and discuss various aspects of preconditioning and multilevel methods. Overall, this editorial serves as a tribute to Professor Axelsson's remarkable influence on numerical analysis and his role as a mentor and collaborator. [Extracted from the article]

This document is an editorial for a special issue of the journal "Mathematical Methods in the Applied Sciences" dedicated to Professor Ioannis Stratis on the occasion of his retirement from the Department of Mathematics at the National and Kapodistrian University of Athens. The articles in this volume were written by invited speakers of a workshop organized in Athens in July 2023. Professor Stratis has had a successful career in the field of Mathematical Analysis and Applied Mathematics, making significant contributions to various areas such as complex media, electromagnetics, mechanics, inverse problems, and mathematical biology. He is also recognized for his teaching and mentoring abilities, as well as his kindness and collaborative nature. The authors express their gratitude for his friendship and wish him a happy and productive retirement. [Extracted from the article]

We derive an asymptotic expansion for the log‐likelihood of Gaussian mixture models (GMMs) with equal covariance matrices in the low signal‐to‐noise regime. The expansion reveals an intimate connection between two types of algorithms for parameter estimation: the method of moments and likelihood optimizing algorithms such as Expectation‐Maximization (EM). We show that likelihood optimization in the low SNR regime reduces to a sequence of least squares optimization problems that match the moments of the estimate to the ground truth moments one by one. This connection is a stepping stone towards the analysis of EM and maximum likelihood estimation in a wide range of models. A motivating application for the study of low SNR mixture models is cryo‐electron microscopy data, which can be modeled as a GMM with algebraic constraints imposed on the mixture centers. We discuss the application of our expansion to algebraically constrained GMMs, among other example models of interest. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

This paper deals with a fourth‐order parabolic equation involving variable exponents, subject to Navier boundary conditions. We firstly give an optimal classification on the initial energy for blow‐up or global solutions, which is characterized by the sign of the Nehari energy and the difference between the initial energy and the depth of the potential well. Then, large time estimates of solutions are obtained. The results of the paper extend and complete the ones in "Applied Mathematics Letters 78(2018)141–146." [ABSTRACT FROM AUTHOR]

Let Ω⊂ℝn be a C1 domain, or more generally, a Lipschitz domain with small local Lipschitz constant. In this paper it is shown that if u is a function harmonic in Ω and continuous in Ω¯, which vanishes in a relatively open subset ∑⊂∂Ω; moreover, the normal derivative ∂νu vanishes in a subset of ∑ with positive surface measure; then u is identically zero. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

We construct infinitely many admissible weak solutions to the 2D incompressible Euler equations for vortex sheet initial data. Our initial datum has vorticity concentrated on a simple closed curve in a suitable Hölder space and the vorticity may not have a distinguished sign. Our solutions are obtained by means of convex integration; they are smooth outside a "turbulence" zone which grows linearly in time around the vortex sheet. As a by‐product, this approach shows how the growth of the turbulence zone is controlled by the local energy inequality and measures the maximal initial dissipation rate in terms of the vortex sheet strength. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

Summary: This special issue on the Pervasive Nature of HPC (PN‐HPC) collects an extension of the most valuable works presented at the sixth Workshop on Models, Algorithms and Methodologies for Hybrid Parallelism in New HPC Systems (MAMHYP‐22), held in Gdansk (Poland) in September 2022, jointly with the 14th conference on Parallel Processing and Applied Mathematics (PPAM‐22). New original papers related to the workshop themes are also included. The final aim is to provide a glimpse of the current state of knowledge related to the development of efficient methodologies and algorithms for HPC systems with multiple forms of parallelism. [ABSTRACT FROM AUTHOR]

We investigate the relationship between the N‐clock model (also known as planar Potts model or ℤN‐model) and the XY model (at zero temperature) through a Γ‐convergence analysis of a suitable rescaling of the energy as both the number of particles and N diverge. We prove the existence of rates of divergence of N for which the continuum limits of the two models differ. With the aid of Cartesian currents we show that the asymptotics of the N‐clock model in this regime features an energy that may concentrate on geometric objects of various dimensions. This energy prevails over the usual vortex‐vortex interaction energy. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

This document is an editorial on the special issue of Mathematical Methods in the Applied Sciences (MMAS) for the International Conference of Numerical Analysis and Applied Mathematics 2021 (ICNAAM 2021). The special issue includes the best papers presented at ICNAAM 2021, which were selected through an international peer-review process. The author expresses gratitude to the Editor-in-Chief of MMAS, the anonymous reviewers, and Wiley for their contributions to the special issue. The document concludes by inviting readers to enjoy reading the special issue and find inspiration for their own research. [Extracted from the article]

Perturbation bounds for singular subspaces have a wide range of applications, including high‐dimensional statistics, machine learning, and applied mathematics. This paper focuses on the perturbation bounds of singular subspaces under a signal matrix interfered by a special heteroscedastic noise matrix in which its same row or column shares similar variance. First of all, we extend homoscedastic results of T. Tony Cai and Anru Zhang (see Rate‐optimal perturbation bounds for singular subspaces with applications to high‐dimensional statistics, The Annals of Statistics, 2018, 46(1), 60–89) to heteroscedastic cases. Then we apply the developed tools to the heteroskedastic clustering model. We find out that our upper bound of clustering misclassification rate is better than the one of T. Tony Cai, Rungang Han, and Anru Zhang (see arXiv:2008.12434, 2020). [ABSTRACT FROM AUTHOR]

Motivated by the work of Colin de Verdière and Saint‐Raymond on spectral theory for zeroth‐order pseudodifferential operators on tori, we consider viscosity limits in which zeroth‐order operators, P, are replaced by P + iν Δ, ν > 0. By adapting the Helffer–Sjöstrand theory of scattering resonances, we show that, in a complex neighbourhood of the continuous spectrum, eigenvalues of P + iν Δ have limits as the viscosity ν goes to 0. In the simplified setting of tori, this justifies claims made in the physics literature. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

Our first main result is that correlations between monomers in the dimer model in ℤd do not decay to 0 when d>2. This is the first rigorous result about correlations in the dimer model in dimensions greater than 2 and shows that the model behaves drastically differently than in two dimensions, in which case it is integrable and correlations are known to decay to zero polynomially. Such a result is implied by our more general, second main result, which states the occurrence of a phase transition in the model of lattice permutations, which is related to the quantum Bose gas. More precisely, we consider a self‐avoiding walk interacting with lattice permutations and we prove that, in the regime of fully packed loops, such a walk is 'long' and the distance between its endpoints grows linearly with the diameter of the box. These results follow from the derivation of a version of the infrared bound from a new general probabilistic settings, with coloured loops and walks interacting at sites and walks entering into the system from some 'virtual' vertices. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

Metaphysics faces a threat from apparently metaphysics‐friendly non‐epistemic forms of semantics, on which sentences express "worldly" propositions—for example, functions from worlds to truth‐values. The threat goes back to Wittgenstein"s Tractatus Logico‐Philosophicus and is pressed in different forms by various contemporary philosophers. It is that metaphysical claims turn out either trivially true or trivially false, because they express the same proposition as a tautology or contradiction. The problem is shown to generalize to accounts on which sentences express Russellian structured propositions. It applies to logic and mathematics as well as metaphysics. Attempts to solve it by reinterpreting apparently non‐contingent claims as contingent metalinguistic claims or by invoking Fregean semantics are shown to fail. The underlying problem concerns necessary equivalence, not necessary truth, and arises in all domains. To solve it, we must recognize that the form of our representations plays an ineliminable cognitive role that cannot be reduced to their content. [ABSTRACT FROM AUTHOR]

This article is an editorial on a special issue of the journal "Mathematical Methods in the Applied Sciences" titled "Physically Relevant Moving Boundary Problems." The special issue focuses on moving boundary problems that are encountered in various research fields, including physical and chemical processes, biophysics, and life science. The issue contains 11 original contributions that explore the mathematical theory, models, computational algorithms, and analysis of these moving boundary problems. The authors express gratitude to the reviewers and the entire editorial board for their support and assistance. The authors of the editorial are Irina Nizovtseva and Dmitri Alexandrov. [Extracted from the article]

This document is an erratum statement for the special issue on Integrable Systems and Their Applications in Honor of A. S. Fokas in the journal Studies in Applied Mathematics. The article type "Original Article" has been updated to "Special Issue Article" and the special issue title has been added with a link to the virtual issue. The document also includes a list of articles published in the special issue, along with their authors and DOI links. [Extracted from the article]

Multiple‐choice (MC) items are widely used in educational tests. Distractor analysis, an important procedure for checking the utility of response options within an MC item, can be readily implemented in the framework of item response theory (IRT). Although random guessing is a popular behavior of test‐takers when answering MC items, none of the existing IRT models for distractor analysis have considered the influence of random guessing in this process. In this article, we propose a new IRT model to distinguish the influence of random guessing from response option functioning. A brief simulation study was conducted to examine the parameter recovery of the proposed model. To demonstrate its effectiveness, the new model was applied to the mathematics tests of the Hong Kong Diploma of Secondary Education Examination (HKDSE) from 2015 to 2019. The results of empirical analyses suggest that the complexity of item contents is a key factor in inducing students' random guessing. The implications and applications of the new model to other testing situations are also discussed. [ABSTRACT FROM AUTHOR]

In both theoretical and applied mathematics fields, integral inequalities play a critical role. Due to the behavior of the definition of convexity, both concepts convexity and integral inequality depend on each other. Therefore, the relationship between convexity and integral inequality is strong. By the importance of these concepts, we have introduced the new class of generalized convex function is known as LR‐ h‐convex interval‐valued function (LR‐ h‐convex‐IVF) by means of pseudo‐order relation (≤p). This order relation is defined on interval space. Under the new concept, first, both discrete and continuous new versions of Jensen‐type inequalities are presented by means of pseudo‐order relation. Second, several new Hermite–Hadamard (HH)‐ and Hermite–Hadamard–Fejér (HH‐Fejér)‐type inequalities are also derived for LR‐ h‐convex‐IVFs. Moreover, we have shown that our results include a wide class of new and known inequalities for LR‐ h‐convex‐IVFs and their variant forms as special cases. Useful examples that verify the applicability of the theory developed in this study are presented. It the end, we have proved that the set inclusion "⊆" coincident to pseudo‐order relation "≤p." The concepts and techniques of this paper may be the starting point for further research in this area and used as a tool to investigate the research of probability and optimization, among others. [ABSTRACT FROM AUTHOR]

Summary: This work gives an overview of an innovative class of methods for the analysis of spatial and of functional data observed over complicated two‐dimensional domains. This class is based on regression with regularising terms involving partial differential equations. The associated estimation problems are solved resorting to advanced numerical analysis techniques. The synergical interplay of approaches from statistics, applied mathematics and engineering endows the methods with important advantages with respect to the available techniques, and makes them able to accurately deal with data structures for which the classical techniques are unfit. Spatial regression with differential regularisation is illustrated via applications to the analysis of eco‐colour doppler measurements of blood‐flow velocity, and to functional magnetic resonance imaging signals associated with neural connectivity in the cerebral cortex. [ABSTRACT FROM AUTHOR]

We study the multitime distribution in a discrete polynuclear growth model or, equivalently, in directed last‐passage percolation with geometric weights. A formula for the joint multitime distribution function is derived in the discrete setting. It takes the form of a multiple contour integral of a block Fredholm determinant. The asymptotic multitime distribution is then computed by taking the appropriate KPZ‐scaling limit of this formula. This distribution is expected to be universal for models in the Kardar‐Parisi‐Zhang universality class. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

This special issue of Studies in Applied Mathematics is dedicated to Professor Harvey Segur on the occasion of his 80th birthday. Harvey is a world‐renowned applied mathematician who has had a great impact on the field of nonlinear waves as a researcher and on the next generation of applied mathematicians as an acclaimed teacher. [ABSTRACT FROM AUTHOR]

We compute the topological simple structure set of closed manifolds that occur as total spaces of flat bundles over lens spaces Sl/(ℤ/p) with fiber Tn for an odd prime p and l ≥ 3 provided that the induced ℤ/p‐action on π1(Tn) = ℤn is free outside the origin. To the best of our knowledge this is the first computation of the structure set of a topological manifold whose fundamental group is not obtained from torsionfree and finite groups using amalgamated and HNN‐extensions. We give a collection of classical surgery invariants such as splitting obstructions and ρ‐invariants that decide whether a simple homotopy equivalence from a closed topological manifold to M is homotopic to a homeomorphism. © 2020 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC [ABSTRACT FROM AUTHOR]

Following van der Poorten, we consider a family of nonlinear maps that are generated from the continued fraction expansion of a function on a hyperelliptic curve of genus g. Using the connection with the classical theory of J‐fractions and orthogonal polynomials, we show that in the simplest case g = 1 this provides a straightforward derivation of Hankel determinant formulae for the terms of a general Somos‐4 sequence, which were found in a particular form by Chang, Hu, and Xin. We extend these formulae to the higher genus case, and prove that generic Hankel determinants in genus 2 satisfy a Somos‐8 relation. Moreover, for all g we show that the iteration for the continued fraction expansion is equivalent to a discrete Lax pair with a natural Poisson structure, and the associated nonlinear map is a discrete integrable system. © 2020 the Authors. Communications on Pure and Applied Mathematics is published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

The main result of this paper gives a new construction of extremal Kähler metrics on the total space of certain holomorphic submersions, giving a vast generalisation and unification of results of Hong, Fine and others. The principal new ingredient is a novel geometric partial differential equation on such fibrations, which we call the optimal symplectic connection equation. We begin with a smooth fibration for which all fibres admit a constant scalar curvature Kähler metric. When the fibres admit automorphisms, such metrics are not unique in general, but rather are unique up to the action of the automorphism group of each fibre. We define an equation which, at least conjecturally, determines a canonical choice of constant scalar curvature Kähler metric on each fibre. When the fibration is a projective bundle, this equation specialises to asking that the hermitian metric determining the fibrewise Fubini‐Study metric is Hermite‐Einstein. Assuming the existence of an optimal symplectic connection and the existence of an appropriate twisted extremal metric on the base of the fibration, we show that the total space of the fibration itself admits an extremal metric for certain polarisations making the fibres small. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

We derive a multiscale generalisation of the Bakry‐Émery criterion for a measure to satisfy a log‐Sobolev inequality. Our criterion relies on the control of an associated PDE well‐known in renormalisation theory: the Polchinski equation. It implies the usual Bakry‐Émery criterion, but we show that it remains effective for measures that are far from log‐concave. Indeed, using our criterion, we prove that the massive continuum sine‐Gordon model with β < 6π satisfies asymptotically optimal log‐Sobolev inequalities for Glauber and Kawasaki dynamics. These dynamics can be seen as singular SPDEs recently constructed via regularity structures, but our results are independent of this theory. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

In this paper, a procedure for the detection of the sources of industrial noise and the evaluation of their distances is introduced. The above method is based on the analysis of acoustic and optical data recorded by an acoustic camera. In order to improve the resolution of the data, interpolation and quasi interpolation algorithms for digital data processing have been used, such as the bilinear, bicubic, and sampling Kantorovich (SK). The experimental tests show that the SK algorithm allows to perform the above task more accurately than the other considered methods. [ABSTRACT FROM AUTHOR]

Consider an infinite cloud of hard spheres sedimenting in a Stokes flow in the whole space ℝd. Despite many contributions in fluid mechanics and applied mathematics, there is so far no rigorous definition of the associated effective sedimentation velocity. Calculations by Caflisch and Luke in dimension d = 3 suggest that the effective velocity is well‐defined for hard spheres distributed according to a weakly correlated and dilute point process, and that the variance of the sedimentation speed is infinite. This constitutes the Caflisch‐Luke paradox. In this contribution, we consider a scalar version of this problem that displays the same difficulties in terms of interaction between the differential operator and the randomness, but is simpler in terms of PDE analysis. For a class of hardcore point processes we rigorously prove that the effective velocity is well‐defined in dimensions d > 2 and that the variance is finite in dimensions d > 4, confirming the formal calculations by Caflisch and Luke, and opening a way to the systematic study of such problems. © 2020 Wiley Periodicals LLC [ABSTRACT FROM AUTHOR]

We introduce and study a mean‐field model for a system of spatially distributed players interacting through an evolutionary game driven by a replicator dynamics. Strategies evolve by a replicator dynamics influenced by the position and the interaction between different players and return a feedback on the velocity field guiding their motion. One of the main novelties of our approach concerns the description of the whole system, which can be represent‐dimensional state space (pairs (x, σ) of position and distribution of strategies). We provide a Lagrangian and a Eulerian description of the evolution, and we prove their equivalence, together with existence, uniqueness, and stability of the solution. As a byproduct of the stability result, we also obtain convergence of the finite agents model to our mean‐field formulation, when the number N of the players goes to infinity, and the initial discrete distribution of positions and strategies converge. To this aim we develop some basic functional analytic tools to deal with interaction dynamics and continuity equations in Banach spaces that could be of independent interest. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

For M a simple surface, the nonlinear statistical inverse problem of recovering a matrix field Φ:M→son from discrete, noisy measurements of the SO(n)‐valued scattering data CΦ of a solution of a matrix ODE is considered (n ≥ 2). Injectivity of the map Φ ↦ CΦ was established by Paternain, Salo, and Uhlmann in 2012. A statistical algorithm for the solution of this inverse problem based on Gaussian process priors is proposed, and it is shown how it can be implemented by infinite‐dimensional MCMC methods. It is further shown that as the number N of measurements of point evaluations of CΦ increases, the statistical error in the recovery of Φ converges to 0 in L2(M)‐distance at a rate that is algebraic in 1/N and approaches 1/N for smooth matrix fields Φ. The proof relies, among other things, on a new stability estimate for the inverse map CΦ → Φ. Key applications of our results are discussed in the case n = 3 to polarimetric neutron tomography. © 2020 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC [ABSTRACT FROM AUTHOR]

We continue our study of 'no‐dimension' analogues of basic theorems in combinatorial and convex geometry in Banach spaces. We generalize some results of the paper (Adiprasito, Bárány and Mustafa, 'Theorems of Carathéodory, Helly, and Tverberg without dimension', Proceedings of the Thirtieth Annual ACM‐SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, San Diego, California, 2019) 2350–2360) and prove no‐dimension versions of the colored Tverberg theorem, the selection lemma and the weak ε‐net theorem in Banach spaces of type p>1. To prove these results, we use the original ideas of Adiprasito, Bárány and Mustafa for the Euclidean case, our no‐dimension version of the Radon theorem and slightly modified version of the celebrated Maurey lemma. [ABSTRACT FROM AUTHOR]

Let (M, g) be an asymptotically flat Riemannian 3‐manifold with nonnegative scalar curvature and positive mass. We show that each leaf of the canonical foliation of the end of (M, g) through stable constant mean curvature spheres encloses more volume than any other surface of the same area. Unlike all previous characterizations of large solutions of the isoperimetric problem, we need no asymptotic symmetry assumptions beyond the optimal conditions for the positive mass theorem. This generality includes examples where global uniqueness of the leaves of the canonical foliation as stable constant mean curvature spheres fails dramatically. Our results here resolve a question of G. Huisken on the isoperimetric content of the positive mass theorem. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

Low‐rank matrix recovery from structured measurements has been a topic of intense study in the last decade and many important problems like matrix completion and blind deconvolution have been formulated in this framework. An important benchmark method to solve these problems is to minimize the nuclear norm, a convex proxy for the rank. A common approach to establish recovery guarantees for this convex program relies on the construction of a so‐called approximate dual certificate. However, this approach provides only limited insight into various respects. Most prominently, the noise bounds exhibit seemingly suboptimal dimension factors. In this paper we take a novel, more geometric viewpoint to analyze both the matrix completion and the blind deconvolution scenario. We find that for both these applications the dimension factors in the noise bounds are not an artifact of the proof, but the problems are intrinsically badly conditioned. We show, however, that bad conditioning only arises for very small noise levels: Under mild assumptions that include many realistic noise levels we derive near‐optimal error estimates for blind deconvolution under adversarial noise. © 2020 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC [ABSTRACT FROM AUTHOR]

In this paper, we study the inverse problem of recovering the spatially varying material properties of a solid polycrystalline object from ultrasonic travel time measurements taken between pairs of points lying on the domain boundary. We consider a medium of constant density in which the orientation of the material's lattice structure varies in a piecewise constant manner, generating locally anisotropic regions in which the wave speed varies according to the incident wave direction and the material's known slowness curve. This particular problem is inspired by current challenges faced by the ultrasonic non‐destructive testing of polycrystalline solids. We model the geometry of the material using Voronoi tessellations and study two simplified inverse problems where we ignore wave refraction. In the first problem, the Voronoi geometry itself and the orientations associated to each region are unknowns. We solve this nonsmooth, nonconvex optimisation problem using a multistart non‐linear least squares method. Good reconstructions are achieved, but the method is shown to be sensitive to the addition of noise. The second problem considers the reconstruction of the orientations on a fixed square mesh. This is a smooth optimisation problem but with a much larger number of degrees of freedom. We prove that the orientations can be determined uniquely given enough boundary measurements and provide a numerical method that is more stable with respect to the addition of noise. [ABSTRACT FROM AUTHOR]

We consider the Fröhlich polaron model in the strong coupling limit. It is well‐known that to leading order the ground state energy is given by the (classical) Pekar energy. In this work, we establish the subleading correction, describing quantum fluctuation about the classical limit. Our proof applies to a model of a confined polaron, where both the electron and the polarization field are restricted to a set of finite volume, with linear size determined by the natural length scale of the Pekar problem. © 2020 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

We study local regularity properties for solutions of linear, nonuniformly elliptic equations. Assuming certain integrability conditions on the coefficient field, we prove local boundedness and Harnack inequality. The assumed integrability assumptions are essentially sharp and improve upon classical results by Trudinger. We then apply the deterministic regularity results to the corrector equation in stochastic homogenization and establish sublinearity of the corrector. © 2019 the Authors. Communications on Pure and Applied Mathematics is published by the Courant Institute of Mathematical Sciences and Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]